Infinity categories with duality and hermitian multiplicative infinite loop space machines

Abstract

We show that any preadditive ∞-category with duality gives rise to a direct sum hermitian K-theory spectrum. This assignment is lax symmetric monoidal, thereby producing $E_{\infty }$-ring spectra from preadditive symmetric monoidal ∞-categories with duality. To have examples of preadditive symmetric monoidal ∞-categories with duality we show that any preadditive symmetric monoidal ∞-category, in which every object admits a dual, carries a canonical duality. Moreover we classify and twist the dualities in various ways and apply our definitions for example to finitely generated projective modules over $E_{\infty }$-ring spectra.